Calculator Quadratic Equation

Quadratic Equation Solver

Results:

Enter coefficients and click 'Solve Equation' to find the roots.

function calculateQuadratic() { var a = parseFloat(document.getElementById("coefficientA").value); var b = parseFloat(document.getElementById("coefficientB").value); var c = parseFloat(document.getElementById("coefficientC").value); var resultOutput = document.getElementById("resultOutput"); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultOutput.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { if (b === 0) { if (c === 0) { resultOutput.innerHTML = "This is an identity (0 = 0), infinitely many solutions."; } else { resultOutput.innerHTML = "This is a contradiction (" + c + " = 0), no solution."; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultOutput.innerHTML = "This is a linear equation (a=0). The root is x = " + x.toFixed(4) + "."; } return; } var discriminant = b * b – 4 * a * c; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultOutput.innerHTML = "The equation has two distinct real roots:x₁ = " + x1.toFixed(4) + "x₂ = " + x2.toFixed(4) + "."; } else if (discriminant === 0) { var x = -b / (2 * a); resultOutput.innerHTML = "The equation has one real root (or two equal real roots):x = " + x.toFixed(4) + "."; } else { // discriminant < 0 var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultOutput.innerHTML = "The equation has two complex roots:x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i."; } }

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • a, b, and c are coefficients, with 'a' not equal to zero.
  • x represents the unknown variable.

Quadratic equations are fundamental in mathematics and appear in various fields, including physics (e.g., projectile motion), engineering (e.g., designing parabolic antennas), economics, and computer graphics. Solving a quadratic equation means finding the values of 'x' that satisfy the equation, also known as the roots or zeros of the equation.

The Quadratic Formula

The most common method for solving quadratic equations is using the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

The term inside the square root, (b² – 4ac), is called the discriminant (often denoted by Δ or D). The value of the discriminant determines the nature of the roots:

  • If discriminant > 0: There are two distinct real roots. This means the parabola (the graph of the quadratic equation) intersects the x-axis at two different points.
  • If discriminant = 0: There is exactly one real root (sometimes called a repeated or double root). The parabola touches the x-axis at exactly one point.
  • If discriminant < 0: There are two complex roots (conjugate pairs). The parabola does not intersect the x-axis.

How to Use This Calculator

Our Quadratic Equation Solver simplifies the process of finding the roots of any quadratic equation. Follow these steps:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for 'a', 'b', and 'c'. Remember, 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.
  2. Enter Values: Input these numerical values into the respective fields in the calculator.
  3. Solve: Click the "Solve Equation" button.
  4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex.

Examples

Let's look at a few examples:

Example 1: Two Distinct Real Roots
Equation: x² – 3x + 2 = 0
Here, a = 1, b = -3, c = 2.
Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1 (which is > 0)
Roots: x₁ = 2, x₂ = 1

Example 2: One Real Root (Repeated)
Equation: x² – 4x + 4 = 0
Here, a = 1, b = -4, c = 4.
Discriminant = (-4)² – 4(1)(4) = 16 – 16 = 0
Root: x = 2

Example 3: Two Complex Roots
Equation: x² + 2x + 5 = 0
Here, a = 1, b = 2, c = 5.
Discriminant = (2)² – 4(1)(5) = 4 – 20 = -16 (which is < 0)
Roots: x₁ = -1 + 2i, x₂ = -1 – 2i

This calculator is a handy tool for students, engineers, and anyone needing to quickly and accurately solve quadratic equations without manual calculation.

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